The cosine geometric distribution with count data modeling

In this research work, a new three-parameter lifetime distribution is introduced and studied. It is called the Harris extended Bilal distribution due to its construction from a mixture of the famous Bilal and Harris distributions, resulting from a branching process. The basic properties, such as the moment generating function, moments, quantile function, and Rényi entropy, are discussed. We show that the hazard rate function has ideal features for modeling increasing, upside-down bathtub, and roller-coaster data sets. In a second part, the Harris extended Bilal model is investigated from a statistical viewpoint. The maximum likelihood estimation is used to estimate the parameters, and a simulation study is carried out. The flexibility of the proposed model in a hydrological data analysis scenario is demonstrated using two practical data sets and compared with important competing models. After that, we establish an acceptance sampling plan that takes advantage of all of the features of the Harris extended Bilal model. The operating characteristic values, the minimum sample size that corresponds to the maximum possible defects, and the minimum ratios of lifetime associated with the producer’s risk are discussed.

A new class of distributions with increasing, decreasing, bathtub-shaped and unimodal hazard rate forms called generalized quadratic hazard rate-power series distribution is proposed. The new distribution is obtained by compounding the generalized quadratic hazard rate and power series distributions. This class of distributions contains several important distributions appeared in the literature, such as generalized quadratic hazard rate-geometric, -Poisson, -logarithmic, -binomial and -negative binomial distributions as special cases. We provide comprehensive mathematical properties of the new distribution. We obtain closed-form expressions for the density function, cumulative distribution function, survival and hazard rate functions, moments, mean residual life, mean past lifetime, order statistics and moments of order statistics; certain characterizations of the proposed distribution are presented as well. The special distributions are studied in some details. The maximum likelihood method is used to estimate the unknown parameters. We propose to use EM algorithm to compute the maximum likelihood estimators of the unknown parameters. It is observed that the proposed EM algorithm can be implemented very easily in practice. One data set has been analyzed for illustrative purposes. It is observed that the proposed model and the EM algorithm work quite well in practice.

Specifying a proper statistical model to represent asymmetric lifetime data with high kurtosis is an open problem. In this paper, the three-parameter, modified, slashed, generalized Rayleigh family of distributions is proposed. Its structural properties are studied: stochastic representation, probability density function, hazard rate function, moments and estimation of parameters via maximum likelihood methods. As merits of our proposal, we highlight as particular cases a plethora of lifetime models, such as Rayleigh, Maxwell, half-normal and chi-square, among others, which are able to accommodate heavy tails. A simulation study and applications to real data sets are included to illustrate the use of our results.

In this paper, a new discrete distribution called Binomial-Discrete Lindley (BDL) distribution is proposed by compounding the binomial and discrete Lindley distributions. Some properties of the distribution are discussed including the moment generating function, moments and hazard rate function. The estimation of distribution parameter is studied by methods of moments, proportions and maximum likelihood. A simulation study is performed to compare the performance of the different estimates in terms of bias and mean square errors. Automobile claim data applications are also presented to see that the new distribution is useful in modelling data.

In this paper, we introduce a new continuous probability distribution which is based on the product of linear and exponential functions having a single parameter and hence, we named it ‘One-parameter Linear-exponential distribution’. Different structural properties such as probability density function, probability distribution function and moment generating function of this distribution have been derived. Moments about origin and hence, the first four central moments of the proposed distribution have been obtained. The hazard rate function and the mean residual life function of the proposed distribution have been discussed, and show its flexibility over Lindley distribution. Estimation of parameters has been discussed by the method of moments as well as the method of maximum likelihood. Goodness of fit has been applied to different data-sets which were used by others. It has been observed that the proposed distribution gives better fit to the most of data-sets of same nature, having variance greater than the mean, than Lindley distribution.

The Weibull distribution is one of the widely known lifetime distribution that has been extensively used for modelling data in reliability and survival analysis. A generalization of both the Marshall-OlkinWeibull distribution and the Weibull truncated negative binomial distribution is introduced and studied in this article. Various distributional properties of the new distribution are derived. Estimation of model parameters using the method of maximum likelihood is discussed. Applications to a real data set is provided to show the flexibility and potentiality of the new distribution over other Weibull models. The first order autoregressive minification process with the new distribution as marginal is also developed. We hope that the new model will serve as a good alternative to other models available in the literature for modeling positive real data in several areas.

The objective of the present study is to develop efficient estimation methods for the use of the GEV distribution for quantile estimation in the presence of nonstationarity. Parameter estimation in the nonstationary GEV model is generally done with the maximum likelihood estimation method (ML). In this work, we develop the generalized maximum likelihood estimation method (GML), in which covariates are incorporated into parameters. A simulation study is carried out to compare the performances of the GML and the ML methods in the case of the stationary GEV model (GEV0), the nonstationary case with a linear dependence of the location parameter on covariates (GEV1), the nonstationary case with a quadratic dependence on covariates (GEV2), and the nonstationary case with linear dependence in both location and scale parameters (GEV11). Simulation results show that the GLM method performs better than the ML method for all studied cases. The nonstationary GEV model is also applied to a case study to illustrate its potential. The case study deals with the annual maximum precipitation at the Randsburg station in California, and the covariate process is taken to be the Southern Index Oscillation.

In this paper, we propose a flexible and general family of distributions based on an original power-exponential transformation approach. We call it the modified generalized-G (MGG) family. The elegance and significance of this family lie in the ability to modify the standard distributions by changing their functional forms without adding new parameters, by compounding two distributions, or by adding one or two shape parameters. The aim of this modification is to provide flexible shapes for the corresponding probability functions. In particular, the distributions of the MGG family can possess increasing, constant, decreasing, “unimodal”, or “bathtub-shaped“ hazard rate functions, which are ideal for fitting several real data sets encountered in applied fields. Some members of the MGG family are proposed for special distributions. Following that, the uniform distribution is chosen as a baseline distribution to yield the modified uniform (MU) distribution with the goal of efficiently modeling measures with bounded values. Some useful key properties of the MU distribution are determined. The estimation of the unknown parameters of the MU model is discussed using seven methods, and then, a simulation study is carried out to explore the performance of the estimates. The flexibility of this model is illustrated by the analysis of two real-life data sets. When compared to fair and well-known competitor models in contemporary literature, better-fitting results are obtained for the new model.

This paper introduces a new generalization of the Pareto distribution using the Marshall-Olkin generator and the method of alpha power transformation. This new model has several desirable properties appropriate for modelling right skewed data. The Authors demonstrate how the hazard rate function and moments are obtained. Moreover, an estimation for the new model parameters is provided, through the application of the maximum likelihood and maximum product spacings methods, as well as the Bayesian estimation. Approximate confidence intervals are obtained by means of an asymptotic property of the maximum likelihood and maximum product spacings methods, while the Bayes credible intervals are found by using the Monte Carlo Markov Chain method under different loss functions. A simulation analysis is conducted to compare the estimation methods. Finally, the application of the proposed new distribution to three real-data examples is presented and its goodness-of-fit is demonstrated. In addition, comparisons to other models are made in order to prove the efficiency of the distribution in question.

ABSTRACTThis paper proposes a novel form of the unit new XLindley distribution, which is derived by incorporating the idea of inverse transformation of the cumulative distribution function. The derived distribution is defined on the positive real line and it exhibits a different range of shapes. We also discuss some important statistical properties of the proposed model, including moments, moment‐generating function, survival and hazard rate functions, quantiles, order statistics, etc., to present a comprehensive theoretical framework. In addition to these, fifteen different estimation methods are employed for conducting the parameter estimation for the new distribution. Besides, a simulation study is performed to evaluate their behaviors using some measures like bias, mean squared error, relative error, and absolute differences. Besides, the usefulness and utility of the newly proposed distribution are checked through real‐life applications via some practical datasets. This practical application is conducted by comparing the new distribution with some other existing counterparts in terms of the value of some model selection criterion.

A flexible distribution has been introduced to handle random variables in the unit interval. This distribution is based on an exponential transformation of the truncated positive normal distribution with two parameters and can effectively fit data with varying degrees of skewness and kurtosis. Therefore, it presents an alternative for modeling this type of data. Several mathematical and statistical properties of this distribution have been derived, such as moments, hazard function, the Bonferroni curve, and entropy. Moreover, we investigate the characterizations of the proposed distribution based on its hazard function. Parameter estimation has been performed using both the maximum likelihood method and method of the moments. Because of this, we were able to determine the best critical region and the information matrix, facilitating the calculation of asymptotic confidence intervals. A simulation study is presented to analyze the behavior of the obtained estimators for different sample sizes. To demonstrate the suitability of the proposed distribution, applications and goodness-of-fit tests have been performed on two practical data sets.

Estimation of the parameters of a type I geometric distribution from truncated observations on conception delays

In this paper, we propose a new bivariate distribution with the Gompertz marginals. Some properties of this new bivariate distribution have been investigated. Several properties of this distribution have been discussed. Parameters estimation using moments and maximum likelihood methods are obtained. A numerical illustration experiments have been performed to see the behavior of the MLEs. One data set has been analyzed for illustrative purpose.

Three-parameter Weibull probability distribution function is used to represent the time-to-failure data. For parameter estimation, the errors-in-variables, maximum likelihood, and least squares methods are compared. The results obtained from five data sets show that maximum likelihood method gives the most reliable parameter estimates which are close to those obtained by errors-in-variables approach. The least squares method gave the poorest results in most cases. For parameter estimation, the Luus-Jaakola direct search optimization procedure yielded the optimal parameter values in negligible computation time, taking less than a second of computation time on a Pentium4/2.66MHz personal computer for 50 data points.

One-inflation in zero-truncated count data has recently found considerable attention. In this regard, zero-truncated Geometric distribution and distribution to a point mass at one are used to create a one-inflated model, namely, one-inflated zero-truncated Geometric distribution. Its reliability characteristics, generating functions, and distributional properties are investigated in detail, which includes survival function, hazard rate function, reverse hazard rate function, probability generating function, characteristic function, variance, skewness, and kurtosis. Monte Carlo simulation have been undertaken to evaluate the effectiveness of the maximum likelihood estimators. To test the compatibility of our proposed model, the baseline model and the proposed model are distinguished by using the two different test procedures. The adaptability of the suggested model is demonstrated using two real-life datasets from separate domains by taking various performance measures into consideration.